Proceedings of the 2015 International Workshop on Parallel Symbolic Computation最新文献_第2页

3-ranks for strongly regular graphs 强正则图为3阶

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation Pub Date : 2015-07-10 DOI: 10.1145/2790282.2790295

A. Novocin, David Saunders, Alexander Stachnik, Bryan S. Youse

{"title":"3-ranks for strongly regular graphs","authors":"A. Novocin, David Saunders, Alexander Stachnik, Bryan S. Youse","doi":"10.1145/2790282.2790295","DOIUrl":"https://doi.org/10.1145/2790282.2790295","url":null,"abstract":"In the study of strongly regular graphs, ranks of adjacency matrices (Laplacians actually) are extensively used to demonstrate inequivalence of graphs. Constructions have been given for several families of graphs. Formulas for the ranks in these families are an important tool for understanding their properties. The first and computational challenge is to compute rank modulo 3 of some very large matrices. To our advantage is that the ranks are expected to be relatively small. Typically in these families, the matrix dimension is 3k while the rank modulo 3 is in the vicinity of 2k. Here we discuss a high performance parallel solution to the problem. It involves parallelism at three levels: word-level vectorization of field elements, shared-memory multi-core, and a multi-node distributed memory and file-system modulated level. The implementation has been applied to the case k = 16, wherein the matrix contains approximately 1.85 peta-entries. The second challenge is to discern a formula for the sequence of ranks in a given graph family. Our computations provide further evidence for an existing conjecture concerning the Dickson family of strongly regular graphs and provide a starting point towards finding a formula for the Ding-Yuan and Cohen-Ganley families of graphs.","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122529252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Parallel sparse interpolation using small primes 使用小素数的并行稀疏插值

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation Pub Date : 2015-06-13 DOI: 10.1145/2790282.2790290

Mohamed Khochtali, Daniel S. Roche, Xisen Tian

引用次数: 2

Gröbner bases over algebraic number fields Gröbner代数数域上的基

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation Pub Date : 2015-04-17 DOI: 10.1145/2790282.2790284

Dereje Kifle Boku, C. Fieker, W. Decker, Andreas Steenpaß

{"title":"Gröbner bases over algebraic number fields","authors":"Dereje Kifle Boku, C. Fieker, W. Decker, Andreas Steenpaß","doi":"10.1145/2790282.2790284","DOIUrl":"https://doi.org/10.1145/2790282.2790284","url":null,"abstract":"Although Buchberger's algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field K = Q(α), a simple extension of Q, where α is an algebraic number, and let f ∈ Q[t] be the minimal polynomial of α. In this paper we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field K. Starting from the ideas of Noro [11], we proceed by joining f to the ideal to be considered, adding t as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2, 3, 10], that is, by inferring information in characteristic zero from information in characteristic p > 0. For suitable primes p, the minimal polynomial f is reducible over Fp. This allows us to apply modular methods once again, on a second level, with respect to the factors of f. The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in Singular [7], outperforms other known methods by far.","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116844111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Accelerating polynomial homotopy continuation on a graphics processing unit with double double and quad double arithmetic 用双双和四双算法在图形处理单元上加速多项式同伦延拓

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation Pub Date : 2015-01-26 DOI: 10.1145/2790282.2790294

J. Verschelde, Xiangcheng Yu

引用次数: 8

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation 2015年并行符号计算国际研讨会论文集

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation Pub Date : 1900-01-01 DOI: 10.1145/2790282

引用次数: 1